Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371007*x^30 + 1582210*x^28 - 5177835*x^26 + 13144625*x^24 - 25995750*x^22 + 39996264*x^20 - 47552155*x^18 + 43139880*x^16 - 29279950*x^14 + 14438100*x^12 - 4952883*x^10 + 1106455*x^8 - 144655*x^6 + 9150*x^4 - 200*x^2 + 1)
gp: K = bnfinit(y^40 - 40*y^38 + 740*y^36 - 8400*y^34 + 65450*y^32 - 371007*y^30 + 1582210*y^28 - 5177835*y^26 + 13144625*y^24 - 25995750*y^22 + 39996264*y^20 - 47552155*y^18 + 43139880*y^16 - 29279950*y^14 + 14438100*y^12 - 4952883*y^10 + 1106455*y^8 - 144655*y^6 + 9150*y^4 - 200*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371007*x^30 + 1582210*x^28 - 5177835*x^26 + 13144625*x^24 - 25995750*x^22 + 39996264*x^20 - 47552155*x^18 + 43139880*x^16 - 29279950*x^14 + 14438100*x^12 - 4952883*x^10 + 1106455*x^8 - 144655*x^6 + 9150*x^4 - 200*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371007*x^30 + 1582210*x^28 - 5177835*x^26 + 13144625*x^24 - 25995750*x^22 + 39996264*x^20 - 47552155*x^18 + 43139880*x^16 - 29279950*x^14 + 14438100*x^12 - 4952883*x^10 + 1106455*x^8 - 144655*x^6 + 9150*x^4 - 200*x^2 + 1)
\( x^{40} - 40 x^{38} + 740 x^{36} - 8400 x^{34} + 65450 x^{32} - 371007 x^{30} + 1582210 x^{28} - 5177835 x^{26} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[40, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(32473210254684090614318847656250000000000000000000000000000000000000000\)
\(\medspace = 2^{40}\cdot 3^{20}\cdot 5^{70}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.91\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2\cdot 3^{1/2}5^{7/4}\approx 57.91460926441345$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(300=2^{2}\cdot 3\cdot 5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(131,·)$, $\chi_{300}(7,·)$, $\chi_{300}(137,·)$, $\chi_{300}(11,·)$, $\chi_{300}(17,·)$, $\chi_{300}(283,·)$, $\chi_{300}(289,·)$, $\chi_{300}(191,·)$, $\chi_{300}(163,·)$, $\chi_{300}(293,·)$, $\chi_{300}(113,·)$, $\chi_{300}(169,·)$, $\chi_{300}(299,·)$, $\chi_{300}(257,·)$, $\chi_{300}(173,·)$, $\chi_{300}(49,·)$, $\chi_{300}(179,·)$, $\chi_{300}(181,·)$, $\chi_{300}(59,·)$, $\chi_{300}(61,·)$, $\chi_{300}(53,·)$, $\chi_{300}(67,·)$, $\chi_{300}(197,·)$, $\chi_{300}(71,·)$, $\chi_{300}(119,·)$, $\chi_{300}(77,·)$, $\chi_{300}(43,·)$, $\chi_{300}(223,·)$, $\chi_{300}(187,·)$, $\chi_{300}(229,·)$, $\chi_{300}(103,·)$, $\chi_{300}(233,·)$, $\chi_{300}(109,·)$, $\chi_{300}(239,·)$, $\chi_{300}(241,·)$, $\chi_{300}(247,·)$, $\chi_{300}(121,·)$, $\chi_{300}(251,·)$, $\chi_{300}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $39$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-3$, $2a^{35}-70a^{33}+1120a^{31}-10850a^{29}+71050a^{27}-332513a^{25}+1146575a^{23}-2959825a^{21}+5754000a^{19}-8405125a^{17}+9133147a^{15}-7244705a^{13}+4070730a^{11}-1546675a^{9}+369275a^{7}-48882a^{5}+2785a^{3}-35a$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-3640210a^{13}+2057510a^{11}-791350a^{9}+193800a^{7}-27132a^{5}+1785a^{3}-35a$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{36}-36a^{34}+594a^{32}-5952a^{30}+40455a^{28}-197316a^{26}+712530a^{24}-1937520a^{22}+3996135a^{20}-6249100a^{18}+7354710a^{16}-6418656a^{14}+4056234a^{12}-1790712a^{10}+523260a^{8}-93024a^{6}+8721a^{4}-324a^{2}+2$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}+31059a^{26}-139230a^{24}+457470a^{22}-1118260a^{20}+2042975a^{18}-2778447a^{16}+2778462a^{14}-1998828a^{12}+999714a^{10}-330120a^{8}+66564a^{6}-7272a^{4}+353a^{2}-4$, $a^{28}-28a^{26}+350a^{24}-2577a^{22}+12419a^{20}-41173a^{18}+96084a^{16}-158780a^{14}+184366a^{12}-147147a^{10}+77506a^{8}-25103a^{6}+4395a^{4}-317a^{2}+4$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a$, $a^{36}-35a^{34}+560a^{32}-5425a^{30}+35525a^{28}-166257a^{26}+573301a^{24}-1480074a^{22}+2878127a^{20}-4207645a^{18}+4582077a^{16}-3654882a^{14}+2082158a^{12}-818454a^{10}+212445a^{8}-34469a^{6}+3171a^{4}-124a^{2}+2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{38}-38a^{36}+665a^{34}-7105a^{32}+51800a^{30}-272831a^{28}+1072043a^{26}-3199600a^{24}+7315449a^{22}-12827353a^{20}+17153675a^{18}-17274040a^{16}+12823100a^{14}-6787523a^{12}+2431726a^{10}-540519a^{8}+62917a^{6}-2330a^{4}-40a^{2}$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58345a^{31}-319207a^{29}+1309378a^{27}-4105764a^{25}+9944675a^{23}-18677724a^{21}+27156492a^{19}-30357308a^{17}+25769774a^{15}-16298205a^{13}+7466758a^{11}-2375309a^{9}+490266a^{7}-58133a^{5}+3070a^{3}-40a$, $a^{37}-37a^{35}+630a^{33}-6545a^{31}+46375a^{29}-237307a^{27}+905814a^{25}-2626649a^{23}+5837952a^{21}-9961645a^{19}+12987202a^{17}-12788029a^{15}+9326863a^{13}-4889184a^{11}+1759120a^{9}-403744a^{7}+52053a^{5}-2909a^{3}+37a$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{37}-37a^{35}+630a^{33}-6545a^{31}+46375a^{29}-237306a^{27}+905787a^{25}-2626326a^{23}+5835698a^{21}-9951480a^{19}+12956195a^{17}-12722970a^{15}+9233277a^{13}-4798951a^{11}+1703505a^{9}-383689a^{7}+48503a^{5}-2710a^{3}+40a$, $a^{27}-27a^{25}+323a^{23}-2254a^{21}+10165a^{19}-31008a^{17}+65076a^{15}-93704a^{13}+90662a^{11}-56485a^{9}+21021a^{7}-4082a^{5}+313a^{3}-4a$, $a^{3}-3a+1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a-1$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a-1$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58345a^{31}-319207a^{29}+1309378a^{27}-4105764a^{25}+9944675a^{23}-18677724a^{21}+27156492a^{19}-30357308a^{17}+25769774a^{15}-16298205a^{13}+7466758a^{11}-2375309a^{9}+490266a^{7}-58133a^{5}+3070a^{3}-39a+1$, $a^{39}-a^{38}-39a^{37}+38a^{36}+702a^{35}-665a^{34}-7735a^{33}+7105a^{32}+58345a^{31}-51800a^{30}-319207a^{29}+272832a^{28}+1309378a^{27}-1072071a^{26}-4105764a^{25}+3199950a^{24}+9944675a^{23}-7318026a^{22}-18677724a^{21}+12839772a^{20}+27156492a^{19}-17194847a^{18}-30357308a^{17}+17370106a^{16}+25769774a^{15}-12981745a^{14}-16298205a^{13}+6971342a^{12}+7466758a^{11}-2577574a^{10}-2375309a^{9}+616189a^{8}+490266a^{7}-86522a^{6}-58133a^{5}+6080a^{4}+3070a^{3}-160a^{2}-39a-1$, $a^{31}-a^{30}-31a^{29}+30a^{28}+434a^{27}-405a^{26}-3627a^{25}+3250a^{24}+20150a^{23}-17250a^{22}-78430a^{21}+63756a^{20}+219603a^{19}-168245a^{18}-447032a^{17}+319770a^{16}+660706a^{15}-436050a^{14}-700245a^{13}+419900a^{12}+518947a^{11}-277134a^{10}-257621a^{9}+119340a^{8}+79704a^{7}-30940a^{6}-13502a^{5}+4200a^{4}+955a^{3}-225a^{2}-12a+2$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58345a^{31}-a^{30}-319207a^{29}+30a^{28}+1309378a^{27}-405a^{26}-4105764a^{25}+3250a^{24}+9944675a^{23}-17250a^{22}-18677724a^{21}+63756a^{20}+27156492a^{19}-168245a^{18}-30357308a^{17}+319770a^{16}+25769774a^{15}-436050a^{14}-16298205a^{13}+419900a^{12}+7466758a^{11}-277134a^{10}-2375309a^{9}+119340a^{8}+490266a^{7}-30940a^{6}-58133a^{5}+4200a^{4}+3070a^{3}-225a^{2}-39a+2$, $a^{6}-6a^{4}-a^{3}+9a^{2}+3a-1$, $a^{37}-37a^{35}+630a^{33}-6545a^{31}-a^{30}+46375a^{29}+30a^{28}-237306a^{27}-405a^{26}+905787a^{25}+3250a^{24}-2626326a^{23}-17250a^{22}+5835698a^{21}+63756a^{20}-9951480a^{19}-168245a^{18}+12956195a^{17}+319770a^{16}-12722970a^{15}-436050a^{14}+9233277a^{13}+419900a^{12}-4798951a^{11}-277134a^{10}+1703505a^{9}+119340a^{8}-383688a^{7}-30940a^{6}+48496a^{5}+4200a^{4}-2696a^{3}-225a^{2}+33a+3$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219603a^{19}-447032a^{17}+660706a^{15}-700245a^{13}+518947a^{11}-257621a^{9}+79704a^{7}-13502a^{5}+955a^{3}-12a-1$, $a^{39}-a^{38}-39a^{37}+39a^{36}+701a^{35}-701a^{34}-7699a^{33}+7699a^{32}+57751a^{31}-57752a^{30}-313256a^{29}+313286a^{28}+1268953a^{27}-1269359a^{26}-3908854a^{25}+3912130a^{24}+9235420a^{23}-9252970a^{22}-16757730a^{21}+16823510a^{20}+23225884a^{19}-23402984a^{18}-24283768a^{17}+24629871a^{16}+18755183a^{15}-19245481a^{14}-10354359a^{13}+10851674a^{12}+3880811a^{11}-4233163a^{10}-905477a^{9}+1072722a^{8}+110602a^{7}-160198a^{6}-3945a^{5}+12051a^{4}-191a^{3}-369a^{2}+4a+3$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58345a^{31}-319206a^{29}+1309349a^{27}-4105387a^{25}+9941775a^{23}-18663051a^{21}+27105155a^{19}-30230235a^{17}+25546070a^{15}-16020800a^{13}+7230677a^{11}-2244023a^{9}+446597a^{7}-50806a^{5}+2640a^{3}-31a-1$, $a^{27}-27a^{25}+323a^{23}-2254a^{21}+10165a^{19}-31008a^{17}+65076a^{15}-93704a^{13}+90662a^{11}-56485a^{9}+21021a^{7}-4082a^{5}+313a^{3}-4a-1$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308944a^{27}-4102137a^{25}+9924525a^{23}-18599295a^{21}+26936910a^{19}-29910465a^{17}+25110020a^{15}-15600900a^{13}+6953543a^{11}-2124683a^{9}+415657a^{7}-46606a^{5}+2415a^{3}-28a-1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576263a^{15}-3640195a^{13}+2057420a^{11}-791075a^{9}+193350a^{7}-26754a^{5}+1645a^{3}-20a-1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-3640210a^{13}+2057510a^{11}-791350a^{9}+193800a^{7}-27133a^{5}+1790a^{3}-40a-1$, $a^{37}-37a^{35}+630a^{33}-6545a^{31}+46375a^{29}-237307a^{27}+905814a^{25}-2626649a^{23}+5837952a^{21}-9961645a^{19}+12987202a^{17}-12788029a^{15}+9326863a^{13}-4889184a^{11}+1759120a^{9}-403744a^{7}+52053a^{5}-2909a^{3}+37a+1$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a-1$, $a^{38}-38a^{36}+665a^{34}-7105a^{32}+51800a^{30}-272831a^{28}+1072043a^{26}-3199600a^{24}+7315449a^{22}-12827353a^{20}+17153675a^{18}-17274040a^{16}+12823100a^{14}+a^{13}-6787523a^{12}-13a^{11}+2431726a^{10}+65a^{9}-540518a^{8}-156a^{7}+62909a^{6}+182a^{5}-2310a^{4}-91a^{3}-56a^{2}+13a+2$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520676a^{11}-260338a^{9}+82212a^{7}-14756a^{5}+1240a^{3}-31a+1$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 31857175264724490000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{40}\cdot(2\pi)^{0}\cdot 31857175264724490000000 \cdot 1}{2\cdot\sqrt{32473210254684090614318847656250000000000000000000000000000000000000000}}\cr\approx \mathstrut & 0.0971884114218969 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371007*x^30 + 1582210*x^28 - 5177835*x^26 + 13144625*x^24 - 25995750*x^22 + 39996264*x^20 - 47552155*x^18 + 43139880*x^16 - 29279950*x^14 + 14438100*x^12 - 4952883*x^10 + 1106455*x^8 - 144655*x^6 + 9150*x^4 - 200*x^2 + 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371007*x^30 + 1582210*x^28 - 5177835*x^26 + 13144625*x^24 - 25995750*x^22 + 39996264*x^20 - 47552155*x^18 + 43139880*x^16 - 29279950*x^14 + 14438100*x^12 - 4952883*x^10 + 1106455*x^8 - 144655*x^6 + 9150*x^4 - 200*x^2 + 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371007*x^30 + 1582210*x^28 - 5177835*x^26 + 13144625*x^24 - 25995750*x^22 + 39996264*x^20 - 47552155*x^18 + 43139880*x^16 - 29279950*x^14 + 14438100*x^12 - 4952883*x^10 + 1106455*x^8 - 144655*x^6 + 9150*x^4 - 200*x^2 + 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 40*x^38 + 740*x^36 - 8400*x^34 + 65450*x^32 - 371007*x^30 + 1582210*x^28 - 5177835*x^26 + 13144625*x^24 - 25995750*x^22 + 39996264*x^20 - 47552155*x^18 + 43139880*x^16 - 29279950*x^14 + 14438100*x^12 - 4952883*x^10 + 1106455*x^8 - 144655*x^6 + 9150*x^4 - 200*x^2 + 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2\times C_{20}$ (as 40T2):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| An abelian group of order 40 |
| The 40 conjugacy class representatives for $C_2\times C_{20}$ |
| Character table for $C_2\times C_{20}$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{10}$ | ${\href{/padicField/11.10.0.1}{10} }^{4}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | $20^{2}$ | $20^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $40$ | $2$ | $20$ | $40$ | |||
|
\(3\)
| Deg $40$ | $2$ | $20$ | $20$ | |||
|
\(5\)
| Deg $40$ | $20$ | $2$ | $70$ |